Why does the quadratic formula give two answers?

Conditions

• The quadratic formula is being applied to solve an equation.
• The expression under the square root (the discriminant) is positive, yielding a real square root.

Reasoning, step by step

1. Observe the $\pm$ symbol in the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
2. Understand that $\pm$ represents two distinct cases: one with $+$ and one with $-$.
3. Split the final fraction into two separate equations.
4. Calculate the first solution using the plus sign.
5. Calculate the second solution using the minus sign.
6. Conclude that the two calculations provide the two roots of the quadratic equation.

Example

In the video, the expression $x = \frac{-3 \pm 5}{4}$ is highlighted. The speaker explains that the $\pm$ symbol means we need to break it up into two parts: $x = \frac{-3 + 5}{4}$ and $x = \frac{-3 - 5}{4}$, which evaluate to $\frac{1}{2}$ and $-2$ respectively.

Common misconceptions

• Assuming the $\pm$ symbol means you should choose either the plus or the minus sign arbitrarily, rather than calculating both.
• Believing that a quadratic equation can only have one solution if the discriminant is a perfect square.

Watch the explanation

Explore next

• What is the role of the expression under the square root in the quadratic formula?
• How do you verify if a calculated root of a quadratic equation is correct?